CN106710245A - Ramp control method for multiple lanes of expressway based on density - Google Patents

Abstract

The invention discloses a ramp control method for multiple lanes of an expressway based on density, which belongs to the technical field of traffic information. Firstly, ramp traffic and mainline traffic are considered, and the density parameters of ramp traffic and mainline traffic are simultaneously integrated in a control target. Thus, the mainline traffic flow is the largest, and the problems of long queue on the ramp and big queuing delay are solved. Secondly, density instead of occupancy and traffic is directly selected as a control parameter, so that convenient data acquisition and unique description of traffic are realized. Thirdly, the difference between multiple lanes in traffic flow characteristic and the lane-changing behaviors of vehicles are fully considered, the traffic state is accurately judged from a more micro perspective, and therefore, the effect of ramp control is improved.

Description

Through street multilane ramp metering rate method based on density

Technical field

The present invention relates to transport information technical field, a kind of through street multilane ring road control based on density is show in particular Method processed.

Background technology

Typical ramp metering rate algorithm ALINEA with main line occupation rate as control parameter, the purpose is to tie up main line occupation rate Hold and expecting near occupation rate, in actual control process, the measure of occupation rate is subject to traffic current density, Vehicle length and inspection The influence of the factors such as test coil length, in the case of mixed traffic flow, the Accurate Determining of occupation rate is restricted.ALINEA's Expansion algorithm UP-ALINEA, FL-ALINEA, UF-ALINEA, AD-ALINEA, AU-ALINEA, PI-ALINEA etc. are in order in reality The facility of Data Collection in the application of border, most of algorithms are maintained at the main line magnitude of traffic flow using main line flow as control parameter Near desired value.But newest achievement in research shows, under the same traffic conditions of same road segment, traffic behavior has not Uniqueness.For example, when main line has the identical magnitude of traffic flow, being both likely to be congestion stream mode, it is also possible to be free flow shape State.Therefore the algorithm with main line flow as control parameter has potential defect.In addition, ALINEA and its expansion algorithm directly or Control targe is indirectly turned to main line magnitude of traffic flow maximum, shortcoming is considered to the length that ring road is queued up, ring road may be caused to arrange Team overflows and traffic above-ground is impacted, or causes ring road traffic delay excessive.Meanwhile, ALINEA and its expansion algorithm are from whole The angle in individual section investigates the traffic behavior in control range, but much researchs show simultaneously, same section diverse location The traffic behavior in track has notable difference, and except this, vehicle lane-changing behavior can also produce considerable influence to traffic flow modes, therefore ALINEA and its expansion algorithm may cause to produce deviation to the judgement of traffic flow modes, so as to have influence on the control of fast road ramp Effect processed.

The content of the invention

Goal of the invention

It is a primary object of the present invention to provide a kind of through street multilane ramp metering rate method based on density, solve existing Deposit that judgement of the ramp metering rate strategy to traffic behavior is more rough, may there is relatively large deviation with actual traffic behavior, it is difficult to Reach the problem of preferable control effect corresponding with traffic behavior.

Technical scheme

A kind of through street multilane ramp metering rate method based on density, comprises the following steps：

S1. multilane dynamic density model is set up

Assume initially that as follows：First, length is divided into N sections for the city expressway of L, and the number of track-lines of each section of through street is Identical；Second, to section i, can only at most there are an Entrance ramp or exit ramp coupled；3rd, Entrance ramp Importing wagon flow can only be imported into main line outermost track, and main line vehicle can only carry out lane-change behavior, standdle carrier between adjacent track The lane-change behavior in road is not allowed to be occurred；

Based on above-mentioned it is assumed that providing the through street dynamic density model in the case of only two tracks, defining fast lane is Track 1, kerb lane is track 2, and as i ≠ j, i.e., no Entrance ramp or exit ramp are connected with section；As i=j, i.e., There is an Entrance ramp to be connected with section, then the dynamic density model of interior kerb lane can be expressed as in the case of two tracks：

In formula：Q (k, i, 1) for section i the time period [kT, (k+1) T] leave fast lane flow into downstream vehicle number with The ratio of T, n_1,2(k, i) changes lane to the vehicle number of kerb lane from fast lane for section i in the time period [kT, (k+1) T]；

Define η_y,l(k, i) is to change lane to the l articles track by the y articles track on time period [tT, (t+1) T] interior section i The ratio of vehicle fleet on vehicle number and track y, then above formula can be write as：

The dynamic density equation of Entrance ramp then can be expressed as：

In formula：q_rampinK () is ratio of the vehicle number with T for entering ring road in the time period [kT, (k+1) T], w is entrance circle Road length；

For the dynamic density model of multilane situation, definition X is total track quantity, and l is the l articles track (l=1,2 ... X), as l=1, most fast lane is expressed as, as l=X, is expressed as outermost track, then multilane dynamic density model can It is expressed as follows；

Fast lane：

Kerb lane：

S2. multilane ramp metering rate model is set up based on multilane dynamic density model

In order to set up the model of multilane ramp metering rate, first it is to be understood that for different tracks dynamic density model it is micro- Divide equation, in the case where section i does not have ring road to be connected (i ≠ j), the change of density depends on the outflow flow q of section i-1 (k, i-1, l) with outflow flow q (k, i, l) of section i, define Δ t=Δ kT, then in the case of i ≠ j, the change of density can It is expressed as：

Above formula is deformed, and the limit is taken to both sides and obtained：

Therefore, the density differential equation in the case of i ≠ j can be write as:

In the case of i=j, it is consistent in the case of the density differential equation of 1≤l of fast lane≤X-1 and i ≠ j, for Kerb lane l=X, the change of density by section i-1 outflow flow q (k, i-1, l), outflow flow q (k, i, l) of section i Inflow flow u (k, i) with Entrance ramp is together decided on, therefore, in the case of i=j, the change of track density is represented by：

Same i ≠ j situations equally deform finding limit to above formula, obtain the density differential equation in the case of i=j：

Accordingly, it is considered to the multilane through street dynamic density differential equation of vehicle lane-changing behavior may be summarized as follows：

The Entrance ramp dynamic density differential equation can be write as：

In view of the characteristics of outside lane utilization ratio has significantly different in through street, in multilane ramp metering rate strategy The situation that to introduce control errors function J (k) different to adjust interior kerb lane utilization rate, control errors function can make main line Density is maintained near desired value value, while reducing the queue length of Entrance ramp, control errors equation is defined as follows:

In formula：ρ_c(i, l) is the expected density value in the l articles track of section i, ρ_rampK () is Entrance ramp in the close of moment kT Angle value, λ (l) is the l articles weighting function in track, λ_rampIt is the weighting function of Entrance ramp, and ∑ λ (l)+λ_ramp=1；

In order that control errors function J (k) are minimum, define that single order homogeneous linear differential equation is as follows, the equation has The property of negative exponential function, it is possible to achieve the dynamic reduction of J (k)；

Then the first derivative of control errors function can be written as：

To sum up, multilane ramp metering rate model is obtained as follows：

Beneficial effects of the present invention are as follows：The present invention is directly by the use of traffic current density as the main target of control, a side With occupation rate equally controlling value can maintain near desired value surface density, while ring road queue length can be made as far as possible It is short, taken into account ring road queue length and through street flow, solve the problems, such as ring road queue length it is long influence traffic above-ground and Ring road is queued up and is delayed excessive problem.On the other hand, traffic flow density data is directly obtained from the data that detection coil is detected Take, it is not necessary to other data such as Vehicle length and detection coil length, occupation rate is calculated not in can avoiding mixed traffic environment Accurate problem so that the data acquisition of whole control process is more prone to.Secondly, the present invention considers multi-lane traffic flow Kinetic characteristic, the traffic stream characteristics different to kerb lane in city expressway are described in detail, and emphasis considers car Influence of the lane-change behavior to traffic flow modes, establishes multilane dynamic density model.Additionally, the present invention is in control process Control errors equation is introduced, arterial traffic current density can so be maintained near desired value, and entering with control OK, error is gradually reduced, so as to realize optimal control targe, while the foundation of control errors equation increased control system Stability and antijamming capability.

Brief description of the drawings

Fig. 1 is the situation schematic diagram (i=j) being connected with section without ring road；

Fig. 2 is the situation schematic diagram (i=j) for having an Entrance ramp to be connected with section；

Fig. 3 is detector and Signal layout scheme schematic diagram；

Fig. 4 is the multilane ramp metering rate flow based on density.

Specific embodiment

Below in conjunction with the accompanying drawing in the embodiment of the present application, the technical scheme in the embodiment of the present application is carried out clear, complete Site preparation is described, it is clear that described embodiment is only some embodiments of the present application, rather than whole embodiments.It is based on Embodiment in the application, it is every other that those of ordinary skill in the art are obtained under the premise of creative work is not made Embodiment, belongs to the scope of the application protection.

A kind of through street multilane ramp metering rate method based on density, the present invention comprising multilane dynamic density model and Two parts of multilane ramp metering rate strategy based on real-time density.Wherein multilane dynamic model is used to describing and determining control Traffic circulation state in region, multilane ramp metering rate strategy then implements ramp metering rate under specific traffic circulation state.

Multilane dynamic density model

Classical dynamic density model M ETA and METANET assumes that length is divided into N sections for the expressway of L, then in t= (k+1) (k+1 i) is represented by traffic flow density p of the T moment in the λ tracks of section i：

Wherein, k=1,2..., are the control moment, and T is controlling cycle duration, and q (k, i) is section i in period [kT, (k+ 1) T] vehicle number and cycle T ratio, λ is lane number.

Tan Manchun etc. is on the basis of META models, it is contemplated that safe distance, speed between vehicle lane-changing behavior and vehicle, The other factors such as Path selection, it is proposed that an Expressway Traffic Flow dynamic discrete model for multilane.

In formula：Rs represents vehicle origin and destination；(t, i, l) represents the l articles track of t section i；C_iRepresent section i's Gather and l ∈ C in track_i；L_iRepresent the length of section i；T represents the sampling period；ρ^rs(t, i, l) represents the l articles track of section i Origin and destination are the averag density of the vehicle in t of rs；u^rs(t, i, l) is represented in the time period [tT, (t+1) T] by Entrance ramp The vehicle number of track (i, l) is imported divided by T；e^rs(t, i, l)] represent time period [tT, (t+1) T] inside lane l through exit ramp from The vehicle number opened is divided by T；q^rs(t, i, l) is represented and track l is left in the time period [tT, (t+1) T], and the vehicle number for flowing into downstream is removed With T；Represent time period [tT, (t+1) T] interior section i it is upper by the y articles track change lane to the l articles vehicle number in track and The ratio of vehicle fleet on the y of track；Track set close with track l in the upstream section i-1 of expression track l,

Although model presented above describes the motion state of traffic flow in reality, terminal in model well Definition increased the difficulty of Data Collection, and from for the angle of ramp metering rate, these data are not required in that.Cause This, the present invention is primarily based on it is assumed hereinafter that being improved multilane dynamic density model.

Length is divided into N sections for the city expressway of L, and the number of track-lines of each section of through street is identical；

To section i, can only at most there are an Entrance ramp or exit ramp coupled；

The remittance wagon flow of Entrance ramp can only be imported into main line outermost track, and main line vehicle can only be in adjacent track Between carry out lane-change behavior, the lane-change behavior in standdle carrier road is not allowed to be occurred.

First, the through street dynamic density model in the case of only two tracks is given.Definition fast lane is track 1, outward Side track is track 2, and as i ≠ j, i.e., no Entrance ramp or exit ramp are connected with section, layout such as Fig. 1 institutes of section i Show；As i=j, that is, there is an Entrance ramp to be connected with section, the layout of section j is as shown in Figure 2 (because the present invention is pin Control to such as mouth ring road, therefore only gived in figure such as the layout scenarios of mouth ring road).

Then the dynamic density model of interior kerb lane can be expressed as in the case of two tracks：

In formula：Q (k, i, 1) for section i the time period [kT, (k+1) T] leave fast lane flow into downstream vehicle number with The ratio of T, n_1,2(k, i) changes lane to the vehicle number of kerb lane from fast lane for section i in the time period [kT, (k+1) T].

Define η_y,l(k, i) is to change lane to the l articles track by the y articles track on time period [tT, (t+1) T] interior section i The ratio of vehicle fleet on vehicle number and track y, then above formula can be write as：

The dynamic density equation of Entrance ramp then can be expressed as：

In formula：q_rampinK () is ratio of the vehicle number with T for entering ring road in the time period [kT, (k+1) T], w is entrance circle Road length.

For the dynamic density model of multilane situation, definition X is total track quantity, and l is the l articles track (l=1,2 ... X), as l=1, most fast lane is expressed as, as l=X, is expressed as outermost track, then multilane dynamic density model can It is expressed as follows.

Fast lane：

Kerb lane：

Multilane ramp metering rate strategy based on real-time density

In order to set up the model of multilane ramp metering rate, first it is to be understood that for different tracks dynamic density model it is micro- Divide equation.In the case where section i does not have ring road to be connected (i ≠ j), the change of density depends on the outflow flow q of section i-1 (k, i-1, l) with outflow flow q (k, i, l) of section i, define Δ t=Δ kT, then in the case of i ≠ j, the change of density can It is expressed as：

Above formula is deformed, and the limit is taken to both sides and obtained：

Therefore, the density differential equation in the case of i ≠ j can be write as:

In the case of i=j, it is consistent in the case of the density differential equation of 1≤l of fast lane≤X-1 and i ≠ j, for Kerb lane l=X, the change of density by section i-1 outflow flow q (k, i-1, l), outflow flow q (k, i, l) of section i Inflow flow u (k, i) with Entrance ramp is together decided on, therefore, in the case of i=j, the change of track density is represented by：

Same i ≠ j situations equally deform finding limit to above formula, obtain the density differential equation in the case of i=j：

Accordingly, it is considered to the multilane through street dynamic density differential equation of vehicle lane-changing behavior may be summarized as follows：

The Entrance ramp dynamic density differential equation can be write as：

In view of the characteristics of outside lane utilization ratio has significantly different in through street, the present invention is in multilane ramp metering rate The situation that control errors function J (k) are introduced in strategy is different to adjust interior kerb lane utilization rate, control errors function can be with Main line density is set to maintain near desired value value, while reducing the queue length of Entrance ramp.Control errors equation is defined as follows:

In formula：ρ_c(i, l) is the expected density value in the l articles track of section i, ρ_rampK () is Entrance ramp in the close of moment kT Angle value, λ (l) is the l articles weighting function in track, λ_rampIt is the weighting function of Entrance ramp, and ∑ λ (l)+λ_ramp=1.

In order that control errors function J (k) are minimum, define that single order homogeneous linear differential equation is as follows, the equation has The property of negative exponential function, it is possible to achieve the dynamic reduction of J (k).

Then the first derivative of control errors function can be written as：

To sum up, multilane ramp metering rate model is obtained as follows：

It is above control principle part, here is implementer's case of the invention.

In order to effectively collect the lane-change behavior of traffic density data and identification main line vehicle, the present invention uses induction coil Data acquisition is carried out with HD video cameras.Wherein, induction coil is used for gathering traffic flow density data, and HD video cameras are used for recognizing control The lane-change behavior of vehicle in region processed.Detector and Signal layout scheme are as shown in Figure 3 (by taking 2 tracks as an example).

The traffic flow data that induction coil and HD video cameras will be collected is transferred to data processing centre and is processed (main Calculating including main line vehicle lane-changing Activity recognition and dynamic density, wherein dynamic density pass through multilane dynamic density model meter Obtain), from obtaining multilane dynamic density data (including ring road dynamic density data and main line dynamic density data).In number According to processing center, by dynamic density data input to the multilane ramp metering rate model based on density, ring road dynamic is just can obtain Regulation rate.After the completion of data processing, dynamic ring road conciliation rate is input to Entrance ramp semaphore by processing center by transmission, just It is capable of achieving dynamic ramp metering rate in real time.Specific control flow chart is as shown in Figure 4.

Claims (1)

1. a kind of through street multilane ramp metering rate method based on density, it is characterised in that comprise the following steps：

S1. multilane dynamic density model is set up

Assume initially that as follows：First, length is divided into N sections for the city expressway of L, and the number of track-lines of each section of through street is identical 's；Second, to section i, can only at most there are an Entrance ramp or exit ramp coupled；3rd, the remittance of Entrance ramp Wagon flow can only be imported into main line outermost track, and main line vehicle can only carry out lane-change behavior between adjacent track, standdle carrier road Lane-change behavior is not allowed to be occurred；

Based on above-mentioned it is assumed that providing the through street dynamic density model in the case of only two tracks, definition fast lane is track 1, kerb lane is track 2, and as i ≠ j, i.e., no Entrance ramp or exit ramp are connected with section；As i=j, that is, have one Individual Entrance ramp is connected with section, then the dynamic density model of interior kerb lane can be expressed as in the case of two tracks：

$\mathrm{\ρ}(k+1,i,1)=\mathrm{\ρ}(k,i,1)+\frac{T}{L}\[q(k,i-1,1)-q(k,i,1)+n_{2,1}(k,i)-n_{1,2}(k,i)\]$

$\left\{\begin{array}{cc}\begin{array}{c}\mathrm{\ρ}(k+1,i,2)=\mathrm{\ρ}(k,i,2)+\frac{T}{L}\[q(k,i-1,2)-q(k,i,2)\\ +n_{1,2}(k,i)-n_{2,1}(k,i)\]\end{array}& i\≠j\\ \begin{array}{c}\mathrm{\ρ}(k+1,i,2)=\mathrm{\ρ}(k,i,2)+\frac{T}{L}\[q(k,i-1,2)-q(k,i,2)\\ +u(k,i)+n_{1,2}(k,i)-n_{2,1}(k,i)\]\end{array}& i=j\end{array}\right.$

In formula：Q (k, i, 1) flows into the vehicle number in downstream with T's for section i leaves fast lane in the time period [kT, (k+1) T] Ratio, n_1,2(k, i) changes lane to the vehicle number of kerb lane from fast lane for section i in the time period [kT, (k+1) T]；

Define η_y,l(k, i) is to change lane to the l articles vehicle in track by the y articles track on time period [tT, (t+1) T] interior section i The ratio with vehicle fleet on the y of track is counted, then above formula can be write as：

$\mathrm{\ρ}(k+1,i,1)=\mathrm{\ρ}(k,i,1)+\frac{T}{L}\[q(k,i-1,1)-q(k,i,1)+{\mathrm{\η}}_{2,1}(k,i)\mathrm{\ρ}(k,i,2)-{\mathrm{\η}}_{1,2}(k,i)\mathrm{\ρ}(k,i,1)$

$\left\{\begin{array}{ccc}\begin{array}{c}\mathrm{\ρ}(k+1,i,2)=\mathrm{\ρ}(k,i,2)+\frac{T}{L}\[q(k,i-1,2)-q(k,i,2)\]\\ +{\mathrm{\η}}_{1,2}(k,i)\mathrm{\ρ}(k,i,1)-{\mathrm{\η}}_{2,1}(k,i)\mathrm{\ρ}(k,i,2)\end{array}& if& i\≠j\\ \begin{array}{c}\mathrm{\ρ}(k+1,i,2)=\mathrm{\ρ}(k,i,2)+\frac{T}{L}\[q(k,i-1,2)-q(k,i,2)+u(k,i)\]\\ +{\mathrm{\η}}_{1,2}(k,i)\mathrm{\ρ}(k,i,1)-{\mathrm{\η}}_{2,1}(k,i)\mathrm{\ρ}(k,i,2)\end{array}& if& i=j\end{array}\right.$

The dynamic density equation of Entrance ramp then can be expressed as：

${\mathrm{\ρ}}_{ramp}(k+1,i)={\mathrm{\ρ}}_{ramp}(k,i)+\frac{T}{w}\[q_{rampin}\left(k\right)-r(k,i)\]$

In formula：q_rampinK () is ratio of the vehicle number with T for entering ring road in the time period [kT, (k+1) T], w is that Entrance ramp is long Degree；

For the dynamic density model of multilane situation, definition X is total track quantity, and l is the l articles track (l=1,2 ... X), As l=1, most fast lane is expressed as, as l=X, is expressed as outermost track, then multilane dynamic density model can table Show as follows；

Fast lane：

$\begin{array}{ccc}\begin{array}{c}\mathrm{\ρ}(k+1,i,l)=\mathrm{\ρ}(k,i,l)+\frac{T}{L}\[q(k,i-1,l)-q(k,i,l)\]\\ +\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{y,l}(k,i)\mathrm{\ρ}(k,i,y)-\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{l,y}(k,i)\mathrm{\ρ}(k,i,l)\end{array}& if& 1\≤l\≤X-1\end{array}$

Kerb lane：

$\left\{\begin{array}{ccc}\begin{array}{c}\mathrm{\ρ}(k+1,i,X)=\mathrm{\ρ}(k,i,X)+\frac{T}{L}\[q(k,i-1,X)-q(k,i,X)\]\\ +\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{y,X}(k,i)\mathrm{\ρ}(k,i,y)-\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{X,y}(k,i)\mathrm{\ρ}(k,i,X)\end{array}& if& i\≠j\\ \begin{array}{c}\mathrm{\ρ}(k+1,i,X)=\mathrm{\ρ}(k,i,X)+\frac{T}{L}\[q(k,i-1,X)-q(k,i,X)+u(k,i)\]\\ +\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{y,X}(k,i)\mathrm{\ρ}(k,i,y)-\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{X,y}(k,i)\mathrm{\ρ}(k,i,X)\end{array}& if& i=j\end{array}\right.$

S2. multilane ramp metering rate model is set up based on multilane dynamic density model

In order to set up the model of multilane ramp metering rate, first it is to be understood that the differential side of the dynamic density model for different tracks Journey, in the case where section i does not have ring road to be connected (i ≠ j), the change of density depends on outflow flow q (k, the i- of section i-1 1, l) with outflow flow q (k, i, l) of section i, define Δ t=Δ kT, then in the case of i ≠ j, the change of density can be represented For：

$\mathrm{\ρ}(k+\mathrm{\Δ}k,i,l)=\mathrm{\ρ}(k,i,l)+\frac{\mathrm{\Δ}t}{L}\[q(k,i-1,l)-q(k,i,l)+\underset{y\∈C_i}{\mathrm{\Σ}}{n}_{y,l}(k,i)-\underset{y\∈C_i}{\mathrm{\Σ}}{n}_{l,y}(k,i)\]$

Above formula is deformed, and the limit is taken to both sides and obtained：

$\underset{\mathrm{\Δ}t=0}{\lim }\frac{\mathrm{\ρ}(k+\mathrm{\Δ}k,i,l)-\mathrm{\ρ}(k,i,l)}{\mathrm{\Δ}t}=\frac{q(k,i-1,l)-q(k,i,l)+\underset{y\∈C_i}{\mathrm{\Σ}}{n}_{y,l}(k,i)-\underset{y\∈C_i}{\mathrm{\Σ}}{n}_{l,y}(k,i)}{L}$

Therefore, the density differential equation in the case of i ≠ j can be write as:

$\stackrel{\·}{\mathrm{\ρ}}(k,i,l)=\frac{d\mathrm{\ρ}(k,i,l)}{dt}=\frac{1}{L}\[q(k,i-1,l)-q(k,i,l)+\underset{y\∈C_i}{\Σ}{n}_{y,l}(k,i)-\underset{y\∈C_i}{\Σ}{n}_{l,y}(k,i)\]$

In the case of i=j, it is consistent in the case of the density differential equation of 1≤l of fast lane≤X-1 and i ≠ j, for outside Track l=X, (k, i-1 l), outflow flow q (k, i, l) of section i and enter by the outflow flow q of section i-1 for the change of density Inflow flow u (k, i) of mouth ring road is together decided on, therefore, in the case of i=j, the change of track density is represented by：

$\mathrm{\ρ}(k+\mathrm{\Δ}k,i,l)=\mathrm{\ρ}(k,i,l)+\frac{\mathrm{\Δ}t}{L}\[q(k,i-1,l)-q(k,i,l)+r(k,i)+\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{y,l}(k,i)-\underset{y\∈C_i}{\mathrm{\Σ}}{\mathrm{\η}}_{l,y}(k,i)\]$

Same i ≠ j situations equally deform finding limit to above formula, obtain the density differential equation in the case of i=j：

$\stackrel{\·}{\mathrm{\ρ}}(k,i,l)=\frac{d\mathrm{\ρ}(k,i,l)}{dt}=\frac{1}{L}\[q(k,i-1,l)-q(k,i,l)+r(k,i)+\underset{y\∈C_i}{\Σ}{n}_{y,l}(k,i)-\underset{y\∈C_i}{\Σ}{n}_{l,y}(k,i)\]$

Accordingly, it is considered to the multilane through street dynamic density differential equation of vehicle lane-changing behavior may be summarized as follows：

The Entrance ramp dynamic density differential equation can be write as：

${\stackrel{\·}{\mathrm{\ρ}}}_{ramp}(k,i)=\frac{1}{w}\[q_{rampin}\left(k\right)-r(k,i)\]$

In view of the characteristics of outside lane utilization ratio has significantly different in through street, introduced in multilane ramp metering rate strategy Control errors function J (k) situations different to adjust interior kerb lane utilization rate, control errors function can make main line density Maintain near desired value value, while reducing the queue length of Entrance ramp, control errors equation is defined as follows:

$J\left(k\right)=\underset{l=1}{\overset{X}{\Σ}}\mathrm{\λ}\left(l\right)|\mathrm{\ρ}(k,i,l)-{\mathrm{\ρ}}_c(i,l)|+{\mathrm{\λ}}_{ramp}{\mathrm{\ρ}}_{ramp}\left(k\right)$

In formula：ρ_c(i, l) is the expected density value in the l articles track of section i, ρ_rampK () is density of the Entrance ramp in moment kT Value, λ (l) is the l articles weighting function in track, λ_rampIt is the weighting function of Entrance ramp, and ∑ λ (l)+λ_ramp=1；

In order that control errors function J (k) are minimum, define that single order homogeneous linear differential equation is as follows, the equation has negative finger The property of number function, it is possible to achieve the dynamic reduction of J (k)；

$pJ\left(k\right)+\stackrel{\·}{J}\left(k\right)=0,J\left(k\right)=J\left(0\right)e^{-pkT}$

Then the first derivative of control errors function can be written as：

$\stackrel{\·}{J}\left(k\right)=\underset{l=1}{\overset{X}{\Σ}}\mathrm{\λ}\left(l\right)\·\stackrel{\·}{\mathrm{\ρ}}(k,i,l)+{\mathrm{\λ}}_{ramp}{\stackrel{\·}{\mathrm{\ρ}}}_{ramp}\left(k\right)$

To sum up, multilane ramp metering rate model is obtained as follows：

$\begin{array}{c}r(k,i)=\frac{L\·w}{\mathrm{\λ}\left(X\right)\·w-{\mathrm{\λ}}_{ramp}\·L}\{\stackrel{\·}{J}\left(k\right)-\frac{1}{L}\underset{l=1}{\overset{X}{\Σ}}\mathrm{\λ}\left(l\right)\·\[q(k,i-1,l)-q(k,i,l)\]\\ -\frac{1}{L}\underset{l=1}{\overset{X-1}{\mathrm{\Σ}}}\[\mathrm{\λ}\left(l\right)-\mathrm{\λ}(l+1)\]\·\[{\mathrm{\η}}_{l+1,l}(k,i)\·\mathrm{\ρ}(k,i,l+1)-{\mathrm{\η}}_{l,l+1}(k,i)\·\mathrm{\ρ}(k,i,l)\]-\frac{{\mathrm{\λ}}_{ramp}\·{\mathrm{\ρ}}_{rampin}\left(k\right)}{w}\}\end{array}.$